For sure movies that were recorded.
It depends on what you mean by "install my new set top box."
If you mean replacing a DVR box with an entirely new box, then you will loose all content recorded on that box's hard disk. Purchased VOD content is stored on Verizon's servers and will not be lost.
It may be possible to move the recoded content to an external hard disk on the old box and be able to authorize the external drive it on the new box. I don't remember if this is possible, but a search of these forums should find an answer as this is a common question.
If you are adding a set top box to your home, then you won't loose anything.
External hard drive is mated to a particular box. Change the box, lose the contents.